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Chapter 3: Problem 27

Factor the polynomial completely and find all its zeros. State the multiplicity of each zero. $$P(x)=x^{4}+3 x^{2}-4$$

Short Answer

Expert verified
Zeros are \( 1, -1, \pm 2i \), each with multiplicity 1.

Step by step solution

01

Identify the Quadratic Form

Notice that the polynomial \( P(x) = x^4 + 3x^2 - 4 \) resembles a quadratic in form if we let \( u = x^2 \). Thus, the polynomial becomes \( u^2 + 3u - 4 \).
02

Factor the Quadratic Polynomial

Factor the polynomial \( u^2 + 3u - 4 \). We need two numbers that multiply to \(-4\) and add to \(3\). These numbers are \(4\) and \(-1\). The polynomial can thus be factored as \((u + 4)(u - 1)\).
03

Substitute Back for \(x\)

Substitute \( u = x^2 \) back into the factors obtained: \( (x^2 + 4)(x^2 - 1) \).
04

Factor Further

Recognize that \( x^2 - 1 \) is a difference of squares and can be further factored as \((x - 1)(x + 1)\). Thus, the entire expression becomes \((x^2 + 4)(x - 1)(x + 1)\).
05

Find the Zeros of Each Factor

Set each factor equal to zero and solve: 1. \( x^2 + 4 = 0 \) gives \( x = \pm 2i \).2. \( x - 1 = 0 \) gives \( x = 1 \).3. \( x + 1 = 0 \) gives \( x = -1 \).
06

Determine Multiplicities of Zeros

From the factors, we have the zeros: - \( x = \pm 2i \) each with multiplicity 1 (from \( x^2 + 4 \)).- \( x = 1 \) with multiplicity 1 (from \( x - 1 \)).- \( x = -1 \) with multiplicity 1 (from \( x + 1 \)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Form
When tackling polynomials, recognizing a structure similar to a quadratic can simplify the factoring process. A quadratic form in its simplest terms is any expression or equation resembling the typical quadratic polynomial, which is \[ ax^2 + bx + c \].For the given polynomial \( P(x) = x^4 + 3x^2 - 4 \), noticing the quadratic form relies on the substitution \( u = x^2 \). By substituting, the equation morphs into \( u^2 + 3u - 4 \), which strongly resembles a familiar quadratic equation.Breaking down this step:
  • Identify expressions containing even powers of a variable.
  • Replace \( x^{2n} \) terms with a new variable like \( u \) to form a quadratic equation.
Once in quadratic form, the polynomial becomes much easier to factor using known techniques for quadratics.
Difference of Squares
The difference of squares is a crucial concept when factoring polynomials. It refers to expressions of the pattern \( a^2 - b^2 \), which can be factored into \((a - b)(a + b)\).In our problem, once we substitute back and obtain \((x^2 + 4)(x^2 - 1)\), observe that \(x^2 - 1\) is a perfect example of the difference of squares.To break it down further:
  • Recognize the pattern \( a^2 - b^2 \).
  • In \( x^2 - 1 \), consider it as \( (x)^2 - (1)^2 \).
  • Factor it as \((x - 1)(x + 1)\).
The key is identifying this pattern quickly. Factoring a difference of squares can often lead you to the solution faster, as it breaks down complex expressions into simpler multipliers.
Complex Numbers
In solving polynomials, sometimes we encounter factors that yield complex numbers as solutions. This usually happens when a quadratic does not have real solutions, often indicated by a negative under the square root sign in the quadratic formula.In the factor \( x^2 + 4 \), setting it equal to zero results in the equation \( x^2 = -4 \). Solving this results in \( x = \pm 2i \), where \( i \) is the imaginary unit defined as \( \sqrt{-1} \).To summarize:
  • When the discriminant (\( b^2 - 4ac \)) of a quadratic is negative, expect complex solutions.
  • If an expression equals \( -b^2 \), the square root yields \( bi \).
  • Complex roots often appear in conjugate pairs such as \( 2i \) and \( -2i \). These roots are crucial when completely solving quadratics within polynomial expressions.
Understanding complex numbers enhances your calculus toolkit and allows you to find all solutions, broadening the horizon beyond real numbers.

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Most popular questions from this chapter

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{2}-2 x-8}{x}$$

Suppose that the rabbit population on Mr. Jenkins' farm follows the formula $$p(t)=\frac{3000 t}{t+1}$$ where \(t \geq 0\) is the time (in months) since the beginning of the year. (a) Draw a graph of the rabbit population. (b) What eventually happens to the rabbit population?

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{x^{2}+3 x}{x^{2}-x-6}$$

Show that the polynomial does not have any rational zeros. $$P(x)=x^{3}-x-2$$

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$P(x)=2 x^{3}-3 x^{2}-8 x+12$$
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